We consider optimal execution strategies for block market orders placed in a limit order book (LOB). We build on the resilience model proposed by Obizhaeva and Wang (2005) but allow for a general shape of the LOB defined via a given density function. Thus, we can allow for empirically observed LOB shapes and obtain a nonlinear price impact of market orders. We distinguish two possibilities for modeling the resilience of the LOB after a large market order: the exponential recovery of the number of limit orders, i.e., of the volume of the LOB, or the exponential recovery of the bid-ask spread. We consider both of these resilience modes and, in each case, derive explicit optimal execution strategies in discrete time. Applying our results to a block-shaped LOB, we obtain a new closed-form representation for the optimal strategy of a risk-neutral investor, which explicitly solves the recursive scheme given in Obizhaeva and Wang (2005). We also provide some evidence for the robustness of optimal strategies with respect to the choice of the shape function and the resilience-type.

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In the past years it has become evident that stochastic effects in regulatory networks play an important role, leading to an increasing in stochastic modelling attempts. In contrast, metabolic networks involving large numbers of molecules are most often modelled deterministically. Going towards the integration of different model systems, gen-regulatory networks become part of a larger model system including signalling pathways and metabolic networks. Thus, the question arises of how to efficiently and accurately simulation such coupled or hybrid systems. We present an algorithmic approach for the simulation of hybrid stochastic and deterministic reaction models that allows for adaptive step-size integration of the deterministic equations while at the same time accurately tracing the stochastic reaction events. We present a mathematical derivation of the hybrid system on the stochastic process level, and present numerical examples that outline the power of hybrid simulations.Résumé. Au cours des dernières années, il est devenu clair que les effets aléatoires jouaient un rôle important dans les réseaux de régulation, et les modèles employés aujourd'hui pour décrire ces réseaux sont de nature stochastique. En revanche, les réseaux métaboliques, qui mettent en jeu un grand nombre de molécules, sont le plus souvent décrits par des modèles déterministes. Dans la modélisation de systèmes complexes, réseaux régulateurs de gènes, chemins de signaux et réseaux métaboliques sont intégrés dans un même modèle. Se pose alors la question de simuler efficacement et avec précision de tels modèles couplés (on parle aussi de modèles hybrides). Nous présentons ici une approche pour la simulation de modèles de réactions hybrides stochastiques/déterministes permettantà la fois d'avoir recoursà des pas de temps adaptatifs dans l'intégration deséquations déterministes et de simuler précisément les réactions décrites par des processus stochastiques. Des simulations numériques illustrent la puissance de ces simulations hybrides.

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Abstract. This paper presents weak second and third order schemes for the Cox-Ingersoll-Ross (CIR) process, without any restriction on its parameters. At the same time, it gives a general recursive construction method for getting weak second order schemes that extend the one introduced by Ninomiya and Victoir. Combine both these results, this allows us to propose a second order scheme for more general affine diffusions. Simulation examples are given to illustrate the convergence of these schemes on CIR and Heston models.